Chekanov’s dichotomy in contact topology

In this paper we study submanifolds of contact manifolds. The main submanifolds we are interested in are contact coisotropic submanifolds. They can be viewed as analogues to symplectic contact coisotropic submanifolds, and can be defined by the symplectic complement with respect to the symplectic structure $d \alpha \vert_\xi$, the restriction of $d \alpha$ on the contact hyperplane field $\xi$. Based on a correspondence between symplectic and contact coisotropic submanifolds, we can show contact coisotropic submanifolds admit a $C^0$-rigidity, similar to Humilière–Leclercq–Seyfaddini’s coisotropic rigidity on symplectic manifolds in [12]. Moreover, based on Shelukhin’s norm in [20] defined on the contactomorphism group, we define a Chekanov type pseudo-metric on the orbit space of a fixed submanifold of a contact manifold. Moreover, we can show a dichotomy of (non-) degeneracy of this pseudo-metric when the dimension of this fixed submanifold is equal to the one for a Legendrian submanifold. This can be viewed as a contact topology analogue to Chekanov’s dichotomy in [5] of (non-)degeneracy of Chekanov-Hofer’s metric on the orbit space of a Lagrangian submanifold. The proof of our result follows several arguments from [23] and [24].

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Received 14 October 2018

Accepted 30 April 2019

Published 14 December 2020